TKK Dissertations 93 Espoo 2007 MODELLING THE UNBALANCED MAGNETIC PULL IN ECCENTRIC-ROTOR ELECTRICAL MACHINES WITH PARALLEL WINDINGS Doctoral Dissertation Helsinki University of Technology Department of Electrical and Communications Engineering Laboratory of Electromechanics Andrej Burakov
72
Embed
MODELLING THE UNBALANCED MAGNETIC PULL …lib.tkk.fi/Diss/2007/isbn9789512290062/isbn9789512290062.pdf · TKK Dissertations 93 Espoo 2007 MODELLING THE UNBALANCED MAGNETIC PULL IN
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
TKK Dissertations 93Espoo 2007
MODELLING THE UNBALANCED MAGNETIC PULL IN ECCENTRIC-ROTOR ELECTRICAL MACHINES WITH PARALLEL WINDINGSDoctoral Dissertation
Helsinki University of TechnologyDepartment of Electrical and Communications EngineeringLaboratory of Electromechanics
Andrej Burakov
TKK Dissertations 93Espoo 2007
Andrej Burakov
Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Electrical and Communications Engineering for public examination and debate in Auditorium S1 at Helsinki University of Technology (Espoo, Finland) on the 25th of October, 2007, at 12 noon.
Helsinki University of TechnologyDepartment of Electrical and Communications EngineeringLaboratory of Electromechanics
Teknillinen korkeakouluSähkö- ja tietoliikennetekniikan osastoSähkömekaniikan Laboratorio
MODELLING THE UNBALANCED MAGNETIC PULL IN ECCENTRIC-ROTOR ELECTRICAL MACHINES WITH PARALLEL WINDINGSDoctoral Dissertation
Distribution:Helsinki University of TechnologyDepartment of Electrical and Communications EngineeringLaboratory of ElectromechanicsP.O. Box 3000FI - 02015 TKKFINLANDURL: http://www.tkk.fi/Units/Electromechanics/Tel. +358-9-451 2381Fax +358-9-451 2991E-mail: [email protected]
Monograph Article dissertation (summary + original articles)
Department Electrical and Communications Engineering Laboratory Electromechanics Field of research Electrical machines Opponent(s) Dr. David Dorrell Supervisor Prof. Antero Arkkio Instructor ---
Abstract This research work is focused on developing simple parametric models of the unbalanced magnetic pull produced in eccentric-rotor electrical machines. The influence of currents circulating in the parallel paths of the stator winding on the unbalanced magnetic pull is given the main attention. The interaction between these currents and those circulating in the rotor cage/damper winding is also considered. First, a parametric force model for an eccentric-rotor salient-pole synchronous machine is developed. The effects of the parallel stator windings are not considered in this model. Next, a low-order parametric force model is built for electrical machines equipped with parallel stator windings but operating without the rotor cage/damper winding. This force model is applicable to salient-pole synchronous machines as well as to induction motors. And finally, a special force model is developed for electrical machines furnished with parallel paths both in the rotor and stator windings. This model accounts for the equalising currents circulating in the rotor and stator windings and also for the interaction between these currents. This third force model can be applied to a salient-pole synchronous machine and to an induction machine. The parameters of the force models are estimated from the results of numerical simulations applying a soft-computing-based estimation program. All the developed force models with the estimated parameters demonstrate a very good performance in a wide whirling frequency range. The effects of parallel paths in the rotor and stator windings on the unbalanced magnetic pull are investigated numerically. The acquired results reveal that the total unbalanced magnetic pull and its constituents related to the fundamental magnetic field and slotting are strongly affected by the presence of parallel paths in the stator winding. However, unlike the rotor cage, parallel stator windings may instigate anisotropy in the unbalanced magnetic pull. In such cases, the results of the numerical impulse response test may differ significantly from the conventional calculation results. It is also shown that, despite the fact that the number of parallel paths in the stator is often substantially lower than the number of parallel paths in the rotor, parallel stator windings may still provide a more efficient UMP mitigation than the rotor cage/damper winding..
Keywords parallel windings, rotor eccentricity, unbalanced magnetic pull
ISBN (printed) 978-951-22-9005-5 ISSN (printed) 1795-2239
ISBN (pdf) 978-951-22-9006-2 ISSN (pdf) 1795-4584
Language English Number of pages 154 p.
Publisher Helsinki University of Technology, Laboratory of Electromechanics
Print distribution Helsinki University of Technology, Laboratory of Electromechanics
The dissertation can be read at http://lib.tkk.fi/Diss/2007/isbn9789512290062/
4
Preface
This research work was conducted in the Laboratory of Electromechanics at Helsinki University of Technology.
I gratefully acknowledge the invaluable help, guidance and support received throughout the whole course of this project from Professor Antero Arkkio. For discovering the potential in me and maintaining the fruitful and encouraging discussions, I owe my sincere gratitude to Professor Asko Niemenmaa (Head of the Laboratory) and Emeritus Professor Tapani Jokinen. I am thankful to Dr. Asmo Tenhunen and Dr. Timo Holopainen for sharing their profound experience and knowledge of electrical machines with me. I am much obliged to Mr. Ari Haavisto for his invaluable help and assistance in practical matters. I remain grateful to Mrs. Marika Schröder, who helped me in mastering the bureaucratic impediments. I am also greatly indebted to my colleagues and friends who made it possible to keep my spirits high all the way through my postgraduate studies.
It is impossible to overestimate the favourable stimulus I experienced sharing my life with Mr. Gábor Gyöngy. The steal-solid support, relentless encouragement and unsurpassed enthusiasm always showed by Mr. Vasil Denchev is also very much appreciated. Dr. Marian Dumitru Negrea is gratefully acknowledged for his thoughtful and attentive advice.
I would like to express heartfelt gratitude to my deceased grandparents Zinaida and Andrej Fitingovs for their invaluable contribution to developing my personality. My dearly-loved parents are thankfully acknowledged for their kindness, dedication and strong belief in me throughout all my life. I also owe great gratitude to my dearest brothers and sister.
For teaching me to conquer the most difficult challenges, I am genuinely obliged to my close friend, Mr. Aleksandr Miteniov “Mituxa”.
My deepest gratitude I reserve for my beloved wife Marija, for being a vital source of inspiration, encouragement and happiness. It is her patience, love and devotion that made it possible to successfully complete this Dissertation and I wholeheartedly dedicate it to her.
The continuous financial support from the Graduate School in Electrical Engineering is acknowledged gratefully. I also highly appreciate the financial aid granted by Emil Aaltosen Foundation.
List of publications........................................................................................................................................... 6
List of symbols and abbreviations .................................................................................................................. 7 Symbols used and their SI units of measure.................................................................................................. 7 Abbreviations.............................................................................................................................................. 10
1 Introduction................................................................................................................................................. 11 1.1 Background of the study ....................................................................................................................... 11 1.2 Aim of the work .................................................................................................................................... 14 1.3 Scientific contribution of the work........................................................................................................ 14 1.4 Structure of the work............................................................................................................................. 15
2 Overview of the electromagnetic field and force calculation .................................................................. 22 2.1 Electromagnetic field computation ....................................................................................................... 22 2.2 Calculation of electromagnetic force .................................................................................................... 25 2.3 Literature review ................................................................................................................................... 31
2.3.1 Analytical methods ........................................................................................................................ 31 2.3.2 Numerical methods........................................................................................................................ 36 2.3.3 Combining the analytical and numerical methods ........................................................................ 40
2.4 Need for further research ...................................................................................................................... 41 2.5 Conclusions........................................................................................................................................... 42
3 Methods of analysis..................................................................................................................................... 44 3.1 Parametric force model ......................................................................................................................... 46 3.2 Parameter estimation............................................................................................................................. 48 3.3 Conclusions........................................................................................................................................... 48
4 Discussion of the results.............................................................................................................................. 49 4.1 Verification of the force models............................................................................................................ 49 4.2 Influence of parallel stator windings on the UMP constituents............................................................. 53 4.3 Other issues related to the parallel stator windings ............................................................................... 54 4.4 Parallel stator windings vs. rotor cage vs. parallel stator windings + rotor cage................................... 55 4.5 Conclusions........................................................................................................................................... 57
• salient- and non-salient-pole synchronous machines equipped with damper
winding and with parallel stator windings;
• induction motors equipped with rotor cage and parallel stator windings.
Interaction between the currents circulating in the parallel circuits of the rotor and
stator may influence the UMP and alter the shape of its FRF to some extent. Although
being relatively small, the effects of this interaction could still be investigated further in
order to improve the accuracy of the parametric force models.
It was shown that the magnetic field harmonics associated with the fundamental
magnetic field and slotting contribute the most to the total UMP. The parallel stator
windings may considerably suppress these UMP constituents and, hence, reduce the net
UMP.
Parallel stator windings were shown to cause anisotropic UMP behaviour,
especially in electrical machines without the rotor cage (damper winding). These findings
conform well to the results presented by Robinson (1943). The author stated that there is
no damping of the UMP along the line between the two parallel stator windings. Due to
this UMP anisotropy, care should be exercised when applying the numerical impulse
response test to analyse such motors.
Parallel paths in the stator winding may provide a more efficient UMP mitigation
than the rotor cage (damper winding), even if the number of parallel circuits in the stator is
substantially lower than the number of parallel circuits in the rotor. Using parallel
connections in the rotor and stator simultaneously ensures the lowest level of the UMP.
The last statement agrees well with the results presented by Arkkio (1996).
The FEA results acquired in this work were not verified by measurements.
However, the employed finite-element models and numerical techniques for the
electromagnetic force calculation were previously validated by Arkkio et al. (2000), Lantto
et al. (2000) and Tenhunen et al. (2003d). The authors studied the electromagnetic forces
in a 15 kW induction motor with different types of rotor eccentricity using the technique
presented by Coulomb (1983). Antila et al. (1998) used Coulomb’s approach and the
method presented by Arkkio to calculate the electromagnetic forces in the radial active
59
magnetic bearings. In all these contributions, the authors reported a very good agreement
between the computed and measured results.
The force models developed were only tested with induction motors and a salient-
pole synchronous machine. However, the application area of the models could be extended
to other types of rotating electrical machines, especially if their construction is similar to
the construction of the motors studied in this work. Thus, by equating the term Bp,2 to zero,
the force model described by Eq. (3.1) can straightforwardly be applied to a cylindrical-
rotor synchronous machine. In permanent magnet synchronous motors, eccentricity
harmonics can induce eddy-currents in the permanent magnets located on the rotor surface.
The eddy-currents would reduce the magnetic field asymmetry and the resultant UMP. In
this way, the permanent magnets could be viewed as a rotor cage. Thus, the proposed force
models could also be applied to these electrical machines.
Although the developed force models were only tested on the electrical machines
with cylindrical circular rotor whirling, the models are also anticipated to perform well
with other types of rotor eccentricity. According to the results by Tenhunen et al. (2003d),
in electrical motor, the rotor motion of which can be described as the combination of
symmetric conical whirling and cylindrical circular whirling, the resultant electromagnetic
force on the rotor is almost the same as in the case of cylindrical circuit whirling, provided
the radii of symmetric conical whirling and cylindrical circular whirling are equal. Thus,
the developed force models could also be applied to describe the UMP in electrical motors
with such rotor whirling motion.
The rotors of both machines simulated were not skewed. The rotor skewing can
significantly affect the voltages induced in the rotor bars by the magnetic field harmonics,
especially the voltages induced by the magnetic field harmonics with short wavelengths
(e.g., harmonics related to stator and rotor slotting). The voltages induced in the rotor bars
by the eccentricity harmonics, which have relatively long wavelengths, remain virtually
unaltered by the skewing. Voltages induced in the rotor bars by a certain magnetic field
harmonic define the amount of damping of this harmonic produced by the rotor cage. Thus,
it is expected that the damping of slot harmonics by the skewed rotor cage can
substantially differ from the damping produced by the rotor cage with straight bars.
According to the results presented in Publication P6, slot harmonics contribute
significantly to the total unbalanced force. However, the UMP component related to
slotting has very small whirling frequency dependence, in the whirling frequency range
considered. As shown in Publication P6, peaks in the FRF of the electromagnetic force are
60
caused primarily by the eccentricity harmonics. Thus, rotor skewing, by affecting the
voltages induced in the rotor bars by the slot harmonics, can alter the average level of the
FRF of the electromagnetic force. Yet, the location of the peaks in the FRF of the force and
their elevation above the average FRF level are expected to be almost the same as in the
machine with straight rotor bars. Therefore, the force models developed are also expected
to be applicable to the machines with skewed rotors.
61
5 Summary
In this thesis, the UMP caused by the eccentric rotor is investigated in a wide
whirling frequency range. Two common types of electrical machine are considered: cage
induction motor and salient-pole synchronous machine. Special attention is drawn to the
effects of parallel stator windings on the UMP.
Simple analytical models describing the UMP are developed and verified using the
FEA. The parametric force models are built for: salient-pole synchronous machines
without the parallel stator windings; electrical motors with parallel circuits in the stator
only; and electrical machines with parallel paths both in the rotor and stator. The two force
models mentioned last were applied to the induction and synchronous motors. All the
presented models exhibited a very good performance throughout the whole whirling
frequency range considered.
Parallel stator windings, similarly to the rotor cage (damper winding), effectively
reduce the UMP. Especially the UMP constituents related to the fundamental magnetic
field and slotting are strongly affected by the parallel paths in the stator. However, unlike
the rotor cage, the parallel stator windings may instigate anisotropy in the UMP. In such
cases, the results of the numerical impulse response test may differ significantly from the
conventional calculation results.
Despite the fact that the number of parallel circuits in the stator is often
substantially lower than the number of parallel circuits in the rotor, the parallel paths in the
stator winding may still provide a more efficient UMP mitigation than the rotor cage
(damper winding).
When parallel circuits are provided both in the rotor and stator, the smallest amount
of the UMP is expected. However, currents circulating in these parallel paths may interact
with each other, thus affecting the UMP and the shape of its FRF.
62
References
Al-Nuaim, N. A., Toliyat, H. A. 1997. A method for dynamic simulation and detection of dynamic air-gap eccentricity in synchronous machines. Proceedings of the IEEE International Electric Machines and Drives Conference, Milwaukee, USA, pp. MA2 5.1-5.3.
Al-Nuaim, N. A., Toliyat, H. A. 1998. Novel method for modeling dynamic air-gap eccentricity in synchronous machines based on modified winding function theory. IEEE Transactions on Energy Conversion, Vol. 13, No. 2, pp. 156-162.
Antila, M., Lantto, E., Arkkio, A. 1998. Determination of forces and linearised parameters of radial active magnetic bearings by finite element technique. IEEE Transactions on Magnetics, Vol. 34, No. 3, pp. 684-694.
Arkkio, A. 1987. Analysis of induction motors based on the numerical solution of the magnetic field and circuit equations. Acta Polytechnica Scandinavica, Electrical Engineering Series, No. 59, Helsinki, 97 p. Available at: http://lib.tkk.fi/Diss/. ISBN 951-666-250-1. (Doctoral thesis).
Arkkio, A., Lindgren, O. 1994. Unbalanced magnetic pull in a high-speed induction motor with an eccentric rotor. Proceedings of the International Conference on Electrical Machines, September 5-8, 1994, Paris, France, Vol. 1, pp. 53-58.
Arkkio, A. 1996. Unbalanced magnetic pull in cage induction motors – dynamic and static eccentricity. Proceedings of the International Conference on Electrical Machines, September 10-12, 1996, Vigo, Spain, pp. 192-197.
Arkkio, A. 1997. Unbalanced magnetic pull in cage induction motors with asymmetry in rotor structures. Eighth International Conference on Electrical Machines and Drives EMD-97, Conference Publication No. 444, September 1-3, 1997, pp. 36-40.
Arkkio, A., Antila, M., Pokki, K., Simon, A., Lantto, E. 2000. Electromagnetic force on a whirling cage rotor. IEE Proceedings – Electric Power Applications, Vol. 147, No. 5, pp. 353-360.
Bastos, J. P. A., Sadowski, N. 2003. Electromagnetic modeling by finite element methods. Marcel Dekker, New York, 490 p.
Belmans, R., Geysen, W., Jordan, H., Vandenput, A. 1982a. Unbalanced magnetic pull and homopolar flux in three-phase induction motors with eccentric rotors. Proceedings of ICEM’82, pp. 916-921.
Belmans, R., Geysen, W., Jordan, H., Vandenput, A. 1982b. Unbalanced magnetic pull in three-phase two-pole motors with eccentric rotor. Proceedings of International Conference on Electrical Machines – Design and Application, London, pp. 65-69.
Belmans, R., Vandenput, A., Geysen, W. 1985 .Determination of the parameters of the radial vibrations of large electric motors. Symposium on Electromechanics and Industrial
63
Electronics Applied to Manufacturing Processes, September 17-19, 1985, San Felice Circeo, Italy, pp. 113-119.
Belmans, R., Vandenput, A., Geysen, W. 1987. Calculation of the flux density and the unbalanced pull in two pole induction machines. Electrical Engineering (Archiv fur Elektrotechnik), Vol. 70, No. 3, pp 151-161.
Berman, M. 1993. On the reduction of magnetic pull in induction motors with off-centre rotor. Conference Record of the IEEE Industry Applications Society Annual Meeting, October 2-8, 1993, Vol. 1, pp. 343-350.
Binns, K. J., Dye, M. 1973. Identification of principal factors causing unbalanced magnetic pull in cage induction motors, Proceeding IEE, Vol. 120, No. 3, pp. 349-354.
Bossio, G., De Angelo, C., Solsona, J., Garcia, G., Valla, M. I. 2004. A 2-d model of the induction machine: an extension of the modified winding function approach. IEEE Transactions on Energy Conversion, Vol. 19, No. 1, pp. 144-150.
Cai, W., Pillay, P., Reichert, K. 2001. Accurate computation of electromagnetic forces in switched reluctance motors. Proceedings of the Fifth International Conference on Electrical Machines and Systems, ICEMS 2001, August 18-20, 2001, Vol. 2, pp. 1065-1071.
Chari, M. V. K., Silvester, P. P. 1980. Finite elements in electrical and magnetic field problems. J. Wiley & Sons, New York, 219 p.
Chari, M., Konrad, A., Palmo, M., D’Angelo, J. 1982. Three-dimensional vector potential analysis for machine field problems. IEEE Transactions on Magnetics, Vol. 18, No. 2, pp. 436-446.
Coulomb, J. L. 1983. A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness. IEEE Transactions on Magnetics, Vol. 19, No. 6, pp 2514-2519.
Covo, A. 1954. Unbalanced magnetic pull in induction motors with eccentric rotors. AIEE Transactions, Vol. 73, pp. 1421-1425.
DeBortoli, M. J., Salon, S. J., Burow, D. W., Slavik, C. J. 1993. Effects of rotor eccentricity and parallel windings on induction machine behaviour: a study using finite element analysis. IEEE Transactions on Magnetics, Vol. 29, No. 2, pp. 1676-1682.
Dorrell, D. G. 1993. Calculation of unbalanced magnetic pull in cage induction machines. Cambridge: University of Cambridge. 169 p. (Doctoral thesis).
Dorrell, D. G., Smith, A. C. 1994. Calculation of UMP in induction motors with series or parallel winding connections. IEEE Transactions on Energy Conversion, Vol. 9, No. 2, pp. 304-310.
64
Dorrell, D. G. 1995a. The influence of rotor skew on unbalanced magnetic pull in cage induction motors with eccentric rotors. Seventh International Conference on Electrical Machines and Drives, September 11-13, 1995, Conference Publication No. 412, pp. 67-71.
Dorrell, D. G. 1995b. The sources and characteristics of unbalanced magnetic pull in cage induction motors with either static or dynamic rotor eccentricity. Proceedings of Stockholm Power Tech, June 18-22, 1995, Stockholm, Sweden. Vol. Electrical Machines and Drives, pp. 229-234.
Dorrell, D. G. 1996. Calculation of unbalanced magnetic pull in small cage induction motors with skewed rotors and dynamic rotor eccentricity. IEEE Transactions on Energy Conversion, Vol. 11, No. 3, pp. 483-488.
Dorrell, D. G. 1999. Experimental behaviour of unbalanced magnetic pull in 3-phase induction motors with eccentric rotors and the relationship with tooth saturation. IEEE Transactions on Energy Conversion, Vol. 14, No. 3, pp. 304-309.
Dorrell, D. G. 2000. Modelling of non-uniform rotor eccentricity and calculation of unbalanced magnetic pull in a 3-phase cage induction motors. Proceeding of ICEM 2000, August 28-30, 2000, Espoo, Finland, pp. 1820-1824.
Dorrell, D. G., Ooshima, M., Chiba, A. 2003. Force analysis of a buried permanent-magnet bearingless motor. IEEE International Electric Machines and Drives Conference, IEMDC 2003, June 1-4, 2003, Vol. 2, pp. 1091-1097.
Ellison, A. J., Moore, C. J. 1968. Acoustic noise and vibration of rotating electric machines. IEE Proceedings, Vol. 115, No. 11, pp. 1633-1640.
Felippa, C. A. 2001. A historical outline of matrix structural analysis: a play in three acts. Computers and Structures, Vol. 79, No. 14, pp. 1313-1324.
Fisher-Hinnen, J. 1899. Dynamo design. Van Nostrand.
Freise, W., Jordan, H. 1962. Unbalanced magnetic pull in 3-phase a.c. machines. ETZ-A, Vol. 83, No. 9, pp. 299-303; Translation: CE Trans. 7836.
Frohne, H. 1967. The practical importance of unbalanced magnetic pull, possibilities of calculating and damping it. Conti Elektro Berichte, Vol. 13, pp. 81-92. Translation: ERA Transactions 1B2617.
Fruchtenicht, J., Jordan, H., Seinsch, H. O. 1982. Exzentrizitätsfelder als Ursache von Lafinstabilitäten bei Asynchronmaschinen. Teil I und II. Archiv fur Elektrotechnik, Vol. 65, pp. 271-292.
Garrigan, N. R., Soong, W. L., Stephens, C. M., Storace, A., Lipo, T. A. 1999. Radial force characteristics of a switched reluctance machine. Industry Applications Conference, Thirty-fourth IAS Annual Meeting, October 3-7, 1999, Vol. 4, pp. 2250-2258.
Gray, A., Pertsch, J. G. 1918. Critical review of the bibliography on unbalanced magnetic pull in dynamo-electric machines. AIEE Transactions, Vol. 37, Part 2, pp. 1417-1424.
65
Haase, H., Jordan, H., Kovacs, K. P. 1972. Vibratory forces as a result of shaft fluxes with two-pole induction machines. Electrotech. (ETZ), Vol. 93, pp. 485-486. Translation: CEGB CE 7822.
Heller, B., Jokl, A. L. 1969. Tangential forces in squirrel-cage induction motors. IEEE Transactions on Power Apparatus and Systems, Vol. 88, No. 4, pp. 484-492.
Hellmund, R. E. 1907. Series versus parallel windings for a.c. motors. Electrical World, No. 49, pp. 388-389.
Holopainen, T. P. 2004. Electromechanical interaction in rotordynamics of cage induction motors. Espoo: VTT Technical Research Centre of Finland, VTT Publications 543, 64 p. Available at: http://lib.tkk.fi/Diss/. ISBN 951-38-6404-9. (Doctoral thesis).
Holopainen, T. P., Tenhunen, A., Lantto, E., Arkkio, A. 2005a. Unbalanced magnetic pull induced by arbitrary eccentric motion of cage rotor in transient operation. Part 1: Analytical model. Electrical Engineering (Archiv fur Elektrotechnik), Vol. 88, No. 1, pp. 13-24.
Holopainen, T. P., Tenhunen, A., Lantto, E., Arkkio, A. 2005b. Unbalanced magnetic pull induced by arbitrary eccentric motion of cage rotor in transient operation. Part 2: Verification and numerical parameter estimation. Electrical Engineering (Archiv fur Elektrotechnik), Vol. 88, No. 1, pp. 25-34.
Joksimovic, G. M., Durovic, M. D., Obradovic, A. B. 1999. Skew and linear rise of MMF across slot modeling – winding function approach. IEEE Transactions on Energy Conversion, Vol. 14, No. 3, pp. 315-320.
Joksimovic, G. M., Penman, J. 2000. The detection of inter-turn short circuits in the stator windings of operating motors. IEEE Transactions on Industrial Electronics, Vol. 47, No. 5, pp. 1078-1084.
Joksimovic, G. M., Durovic, M. D., Penman, J., Arthur, N. 2000. Dynamic Simulation of Dynamic Eccentricity in Induction Machines – Winding Function Approach. IEEE Transactions on Energy Conversion, Vol. 15, No. 2, pp. 143-148.
Jordan, H., Roder, G., Weis, M. 1967. Under what circumstances may mechanical vibrations of the stator core be expected at supply frequency in four-pole three-phase asynchronous machines? Elecktrie, Vol. 21, No. 3, pp. 91-95; Translation: ERA Trans. 1B2578.
von Kaehne, P. 1963. Unbalanced magnetic pull in rotating electrical machines. Survey of published work. ERA Report, ref.: Z/T 142, 1963, 30 p.
Kim, U., Lieu, D. K. 1998. Magnetic field calculation in permanent magnet motors with rotor eccentricity: without slotting effect. IEEE Transactions on Magnetics, Vol. 34, No. 4, Part 2, pp. 2243-2252.
66
Kovacs, K. P. 1977. Two-pole induction-motor vibrations caused by homopolar alternating fluxes. IEEE Transactions on Power Apparatus and Systems, Vol. PAS-96, No. 4, pp. 1105-1108.
Krondl, M. 1956. Self excited radial vibrations of the rotor of induction machines with parallel paths in the winding. Bull. Assoc. Suisse. Elect., Vol. 47, pp. 581-588.
Kyung-Tae Kim; Kwang-Suk Kim; Sang-Moon Hwang; Tae-Jong Kim; Yoong-Ho Jung. 2001. Comparison of magnetic forces for IPM and SPM motor with rotor eccentricity. IEEE Transactions on Magnetics, Vol. 37, No. 5, Part 1, pp. 3448-3451.
Lantto, E., Arkkio, A., Antila, M., Pokki, K., Simon, A. 2000. Electromagnetic forces caused by cage induction motor. 7-th International Symposium on Magnetic Bearings, August 23-25, 2000, ETH Zurich, pp. 589-594.
Li, J. T., Liu, Z. J., Nay, L. H. 2007. Effect of radial magnetic forces in permanent magnet motors with rotor eccentricity. IEEE Transactions on Magnetics, Vol. 43, No. 6, pp. 2525-2527.
Lipo, T. A. 1987. Theory and control of synchronous machines. University of Winconsin-Madison.
Lundstrom, L., Gustavsson, R., Aidanpaa, J.-O., Dahlback, N., Leijon, M. 2007. Influence on the stability of generator rotors due to radial and tangential magnetic pull force. IET Electric Power Applications, Volume. 1, No. 1, pp. 1-8.
Luo, X., Liao, Y., Toliyat, H. A., El-Antably, A., Lipo, T. A. 1995. Multiple coupled circuit modeling of induction machines. IEEE Transactions on Industry Applications, Vol. 31, No. 2, pp. 311-318.
Luomi, J. 1993. Finite element methods for electrical machines. Lecture notes for a postgraduate course in electrical machines. Chalmers University of Technology, Department of Electrical Machines and Power Electronics, Göteborg.
Martin, H. C., Carey G. F. 1973 . Introduction to finite element analysis – Theory and applications. McGraw-Hill Publ., New York, 386 p. ISBN 0-07-040641-3.
Milimonfared, J., Kelk, H. M., Nandi, S., Minassians, A. D., Toliyat, H. A. 1999. A novel approach for broken-rotor-bar detection in cage induction motors. IEEE Transactions on Industry Applications, Vol. 35, No. 5, pp. 1000-1006.
Mizia, J., Adamiak, K., Eastham, A. R., Dawson, G. E. 1988. Finite element force calculation: Comparison of methods for electric machines. IEEE Transactions on Magnetics, Vol. 24, No. 1, pp. 447-450.
Nandi, S., Toliyat, H. A., Parlos, A. G. 1997. Performance analysis of a single phase induction motor under eccentric conditions. IEEE Industry Applications Society, Annual Meeting, October 5-9, 1997, New Orleans, Louisiana, USA, pp. 174-181.
Nandi, S., Bharadwaj, R. M., Toliyat, H. A., Parlos, A. G. 1998. Performance analysis of a three phase induction motor under mixed eccentricity condition. Proceeding of
67
International Conference on Power Electronic Drives and Energy Systems for Industrial Grows, Vol. 1, December 1-3, 1998, pp. 123-128.
Nandi, S., Ahmed, S., Toliyat, H. A. 2001. Detection of rotor slot and other eccentricity related harmonics in a three phase induction motor with different rotor cages. IEEE Transactions on Energy Conversion, Vol. 16, No. 3, pp. 253-260.
Nandi, S., Toliyat, H. A. 2002. Novel frequency-domain-based technique to detect stator interturn faults in induction machines using stator-induced voltages after switch-off. IEEE Transactions on Industry Applications, Vol. 38, No. 1, pp. 101-109.
Neves, C. G. C., Carlson, R., Sadowski, N., Bastos, J. P. A., Soeiro, N. S., Gerges, S. N. Y. 1998. Vibrational behavior of switched reluctance motors by simulation and experimental procedures. IEEE Transactions on Energy Conversion, Vol. 34, No. 5, pp. 3158-3161.
Perers, R., Lundin, U., Leijon, M. 2007. Saturation effects on unbalanced magnetic pull in a hydroelectric generator with an eccentric rotor. IEEE Transactions on Magnetics, Vol. 43, No. 10, pp 3884-3890.
Reichert, K., Freundl, H., Vogt, W. 1976. The calculation of forces and torques within numerical magnetic field calculation methods. Compumag conference, Oxford 1976, pp. 64-73.
Robinson, R. C. 1943. The calculation of unbalanced magnetic pull in synchronous and induction motors. AIEE Transactions, Vol. 62, pp. 620-624.
Robinson, R. C. 1963. Line frequency magnetic vibrations of A.C. machines. AIEE Transactions on Power Apparatus and Systems, Vol. 81, pp. 675-679.
Rosenberg, E. 1918. Magnetic pull in electric machines. Transactions of the American Institute of Electrical Engineers (AIEE), Vol. 37, Part 2, New York, N.Y., USA, pp. 1425-1469.
Sadowski, N., Lefevre, Y., Lajoie-Mazenc, M., Cros, J. 1992. Finite element torque calculation in electrical machines while considering the movement. IEEE Transactions on Magnetics, Vol. 28, No. 2, pp. 1410-1413.
Salon, S. J., DeBortoli, M. J., Burow, D. W., Slavik, C. J. 1992. Calculation of circulating current between parallel windings in induction motors with eccentric rotors by finite element method. International Conference on Electrical Machines, September 15-17, 1992, Manchester, UK, pp. 371-375.
Salon, S., Sivasubramaniam, K., Ergene, L. T. 2001. The effect of asymmetry on torque in permanent magnet motors. IEEE International Electric Machines and Drives Conference, IEMDC 2001, pp. 208-217.
Schlensok, C., Henneberger, G. 2004. Calculation of force excitations in induction machines with centric and excentric positioned rotor using 2-d transient FEM. IEEE Transactions on Magnetics, Vol. 40, No. 2, pp. 782-785.
68
Schuisky, V. W. 1971. Magnetic pull in electrical machines due to eccentricity of the rotor. Electrotech. Masch. Ball, Vol. 88, pp. 391-399. Translation: ERA 2958.
Silvester, P. P., Ferrari, R. L. 1996. Finite elements for electrical engineers. Third Edition, Cambridge University Press, 512 p.
Smith, A. C., Dorrell, D. G. 1996. Calculation and measurement of unbalanced magnetic pull in cage induction motors with eccentric rotors. Part 1: Analytical model. IEE Proceedings – Electric Power Applications, Vol. 143, No. 3, pp. 193-201.
Stavrou, A., Penman, J. 2001. Modelling dynamic eccentricity in smooth air-gap induction machines. IEEE International Electric Machines and Drives Conference, IEMDC 2001, pp. 864-871.
Summers, E. W. 1955. Vibration in 2-pole induction motors related to slip frequency. AIIE Transactions, Vol. 74, pp. 69-72.
Swann, S. A. 1963. Effect of rotor eccentricity on the magnetic field in the air-gap of a non-salient-pole machine. IEE Proceedings, Vol. 110, No. 5, pp. 903-915.
Tarnhuvud, T., Reichert, K. 1988. Accuracy problems of force and torque calculation in FE-systems. IEEE Transactions on Magnetics, Vol. 24, No. 1, pp. 443-446.
Tenhunen, A. 2001. Finite-element calculation of unbalanced magnetic pull and circulating current between parallel windings in induction motor with non-uniform eccentric rotor. Proceedings of Electromotion’01, June 19-20, 2001, Bologna, Italy, pp. 19-24. Available at: http://lib.tkk.fi/Diss/2003/isbn9512266830/.
Tenhunen, A. 2003. Electromagnetic forces acting between the stator and eccentric cage rotor. Espoo: Helsinki University of Technology. (Laboratory of Electromechanics, Report series No. 69, 40 p.). Available at: http://lib.tkk.fi/Diss/. ISBN 951-22-6682-2. (Doctoral thesis).
Tenhunen, A. 2005. Calculation of eccentricity harmonics of the air-gap flux density in induction machines by impulse method. IEEE Transactions on Magnetics, Vol. 41, No. 5, pp. 1904-1907.
Tenhunen, A., Holopainen, T. P., Arkkio, A. 2003a. Impulse method to calculate the frequency response of the electromagnetic forces on whirling cage rotors. IEE Proceedings – Electric Power Applications, Vol. 150, No. 6, pp. 752-756.
Tenhunen, A., Holopainen, T. P., Arkkio, A. 2003b. Effects of equalizing currents on electromagnetic forces of whirling cage rotor. The IEEE International Electric Machines and Drives Conference IEMDC’03, June 1-4, 2003, Vol. 1, pp. 257-263.
Tenhunen, A., Benedetti, T., Holopainen, T. P., Arkkio, A. 2003c. Electromagnetic forces in cage induction motors with rotor eccentricity. The IEEE International Electric Machines and Drives Conference IEMDC’03, June 1-4, 2003, Vol. 3, pp. 1616-1622.
69
Tenhunen, A., Benedetti, T., Holopainen, T. P., Arkkio, A. 2003d. Electromagnetic forces of the cage rotor in conical whirling motion. IEE Proceedings - Electric Power Applications, Vol. 150, No. 5, pp. 563-568.
Tenhunen, A., Holopainen, T. P., Arkkio, A. 2004. Effects of saturation on the forces in induction motors with whirling cage rotor. IEEE Transactions on Magnetics, Vol. 40, No. 2, pp. 766-769.
Timar, P. L. 1989. Noise and vibration of electrical machines. Elsevier Science Publishers, Amsterdam – Oxford – New York – Tokyo, 339 p., ISBN 0-444-98896-3.
Toliyat, H. A., Al-Nuaim, N. A. 1997. Simulation and detection of dynamic air-gap eccentricity in salient pole synchronous machine. IEEE Industry Applications Society Annual Meeting, October 5-9, 1997, New Orleans, USA, pp. 255-262.
Toliyat, H. A., Lipo, T. A. 1995. Transient analysis of cage induction machines under stator, rotor bar and end ring faults. IEEE Transactions on Energy Conversion, Vol. 10, No. 2, pp. 241-247.
Vandevelde, L., Melkebeek, J. A. A. 1994. Theoretical and experimental study of radial forces in relation to magnetic noise of induction motors. International Conference on Electrical Machines ICEM-1994, 3, pp. 419-424.
Wignall, A. N., Gilbert, A. J., Yang, S. J. 1988. Calculation of forces on magnetised ferrous cores using the Maxwell stress method. IEEE Transactions on Magnetics, Vol. 24, No. 1, pp. 459-462.
Williamson, S. 1983. Power factor improvement in cage rotor induction motors. IEE Proceedings – Electric Power Applications, Vol. 130, No. 2, Part B, pp. 121-129.
Williamson, S., Smith, A. C. 1982. Steady-state analysis of 3-phase cage motors with rotor-bar and end-ring faults. IEE Proceedings – Electric Power Applications, Vol. 129, No. 3, Part B, pp. 93-100.
Williamson, S., Mirzoian, K. 1985. Analysis of cage induction motors with stator winding faults. IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No. 7, pp. 1838-1842.
Williamson, S., Abdel-Magied, M. A. S. 1987. Steady-state analysis of double-cage induction motors with rotor-cage faults. IEE Proceedings – Electric Power Applications, Vol. 134, No. 4, Part B, pp. 199-206.
Williamson, S., Adams, N. K. 1989. Cage induction motors with inter-rings. IEE Proceedings – Electric Power Applications, Vol. 136, No. 6, Part B, pp. 263-274.
Yang, S. J. 1975. Acoustic noise from small 2-pole single-phase induction machines. Proceeding IEE, Vol. 122, pp. 1391-1396.
Zhu, Z. Q., Howe, D. 1997. Effect of rotor eccentricity and magnetic circuit saturation on acoustic noise and vibration of single-phase induction motors. Electric Machines and Power Systems, Vol. 25, pp. 443-457.
70
ISBN 978-951-22-9005-5ISBN 978-951-22-9006-2 (PDF)ISSN 1795-2239ISSN 1795-4584 (PDF)